Minimal superalgebras of weak-$^ \ast$ Dirichlet algebras

Abstract:

Let $A$ be a weak-$*$ Dirichlet algebra in ${L^\infty }(m)$ and let ${H^\infty }(m)$ be the weak-$*$ closure of $A$ in ${L^\infty }(m)$. It may happen that there are minimal weak-$*$ closed subalgebras of ${L^\infty }(m)$ that contain ${H^\infty }(m)$ properly. In this paper it is shown that if there is a minimal, proper, weak-$*$ closed superalgebra of ${H^\infty }(m)$, then, in fact, that algebra is the unique least element in the lattice of all proper weak-$*$ closed superalgebras of ${H^\infty }(m)$.